A topology on quantum logics

Abstract:

A uniform topology ${\tau _M}$ induced by a set $M$ of finite measures on a quantum logic $L$ is studied. If $m$ is a valuation on $L$, the topology ${\tau _m}$ induced by $\{ m\}$ is equivalent to the topology induced by the pseudometric $\rho (a,b) = m(a\Delta b)$. If the set $M$ of measures is large enough, the topology ${\tau _M}$ reflects in some sense the structure of $L$: if $L$ is a continuous geometry and the measures are totally additive, ${\tau _M}$ is weaker than the order topology ${\tau _o}$ on $L$. If $L$ is atomic, ${\tau _M}$ is stronger than ${\tau _o}$. On a separable Hilbert space logic, ${\tau _M}$ coincides with the discrete topology. Some cases are found in which ${\tau _M} = {\tau _o}$.